// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_BICGSTAB_H
#define EIGEN_BICGSTAB_H

namespace Eigen {

namespace internal {

/** \internal Low-level bi conjugate gradient stabilized algorithm
 * \param mat The matrix A
 * \param rhs The right hand side vector b
 * \param x On input and initial solution, on output the computed solution.
 * \param precond A preconditioner being able to efficiently solve for an
 *                approximation of Ax=b (regardless of b)
 * \param iters On input the max number of iteration, on output the number of performed iterations.
 * \param tol_error On input the tolerance error, on output an estimation of the relative error.
 * \return false in the case of numerical issue, for example a break down of BiCGSTAB.
 */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool
bicgstab(const MatrixType& mat,
		 const Rhs& rhs,
		 Dest& x,
		 const Preconditioner& precond,
		 Index& iters,
		 typename Dest::RealScalar& tol_error)
{
	using std::abs;
	using std::sqrt;
	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar, Dynamic, 1> VectorType;
	RealScalar tol = tol_error;
	Index maxIters = iters;

	Index n = mat.cols();
	VectorType r = rhs - mat * x;
	VectorType r0 = r;

	RealScalar r0_sqnorm = r0.squaredNorm();
	RealScalar rhs_sqnorm = rhs.squaredNorm();
	if (rhs_sqnorm == 0) {
		x.setZero();
		return true;
	}
	Scalar rho = 1;
	Scalar alpha = 1;
	Scalar w = 1;

	VectorType v = VectorType::Zero(n), p = VectorType::Zero(n);
	VectorType y(n), z(n);
	VectorType kt(n), ks(n);

	VectorType s(n), t(n);

	RealScalar tol2 = tol * tol * rhs_sqnorm;
	RealScalar eps2 = NumTraits<Scalar>::epsilon() * NumTraits<Scalar>::epsilon();
	Index i = 0;
	Index restarts = 0;

	while (r.squaredNorm() > tol2 && i < maxIters) {
		Scalar rho_old = rho;

		rho = r0.dot(r);
		if (abs(rho) < eps2 * r0_sqnorm) {
			// The new residual vector became too orthogonal to the arbitrarily chosen direction r0
			// Let's restart with a new r0:
			r = rhs - mat * x;
			r0 = r;
			rho = r0_sqnorm = r.squaredNorm();
			if (restarts++ == 0)
				i = 0;
		}
		Scalar beta = (rho / rho_old) * (alpha / w);
		p = r + beta * (p - w * v);

		y = precond.solve(p);

		v.noalias() = mat * y;

		alpha = rho / r0.dot(v);
		s = r - alpha * v;

		z = precond.solve(s);
		t.noalias() = mat * z;

		RealScalar tmp = t.squaredNorm();
		if (tmp > RealScalar(0))
			w = t.dot(s) / tmp;
		else
			w = Scalar(0);
		x += alpha * y + w * z;
		r = s - w * t;
		++i;
	}
	tol_error = sqrt(r.squaredNorm() / rhs_sqnorm);
	iters = i;
	return true;
}

}

template<typename _MatrixType, typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar>>
class BiCGSTAB;

namespace internal {

template<typename _MatrixType, typename _Preconditioner>
struct traits<BiCGSTAB<_MatrixType, _Preconditioner>>
{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
};

}

/** \ingroup IterativeLinearSolvers_Module
 * \brief A bi conjugate gradient stabilized solver for sparse square problems
 *
 * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient
 * stabilized algorithm. The vectors x and b can be either dense or sparse.
 *
 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
 *
 * \implsparsesolverconcept
 *
 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
 * and NumTraits<Scalar>::epsilon() for the tolerance.
 *
 * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
 *
 * \b Performance: when using sparse matrices, best performance is achied for a row-major sparse matrix format.
 * Moreover, in this case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
 * See \ref TopicMultiThreading for details.
 *
 * This class can be used as the direct solver classes. Here is a typical usage example:
 * \include BiCGSTAB_simple.cpp
 *
 * By default the iterations start with x=0 as an initial guess of the solution.
 * One can control the start using the solveWithGuess() method.
 *
 * BiCGSTAB can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
 *
 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
 */
template<typename _MatrixType, typename _Preconditioner>
class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType, _Preconditioner>>
{
	typedef IterativeSolverBase<BiCGSTAB> Base;
	using Base::m_error;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_iterations;
	using Base::matrix;

  public:
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

  public:
	/** Default constructor. */
	BiCGSTAB()
		: Base()
	{
	}

	/** Initialize the solver with matrix \a A for further \c Ax=b solving.
	 *
	 * This constructor is a shortcut for the default constructor followed
	 * by a call to compute().
	 *
	 * \warning this class stores a reference to the matrix A as well as some
	 * precomputed values that depend on it. Therefore, if \a A is changed
	 * this class becomes invalid. Call compute() to update it with the new
	 * matrix A, or modify a copy of A.
	 */
	template<typename MatrixDerived>
	explicit BiCGSTAB(const EigenBase<MatrixDerived>& A)
		: Base(A.derived())
	{
	}

	~BiCGSTAB() {}

	/** \internal */
	template<typename Rhs, typename Dest>
	void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
	{
		m_iterations = Base::maxIterations();
		m_error = Base::m_tolerance;

		bool ret = internal::bicgstab(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error);

		m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
	}

  protected:
};

} // end namespace Eigen

#endif // EIGEN_BICGSTAB_H
